A new multi-attribute decision making approach based on new score function and hybrid weighted score measure in interval-valued Fermatean fuzzy environment

Interval-valued Fermatean fuzzy sets (IVFFSs) were introduced as a more effective mathematical tool for handling uncertain information in 2021. In this paper, firstly, a novel score function (SCF) is proposed based on IVFFNs that can distinguish between any two IVFFNs. And then, the novel SCF and hybrid weighted score measure were used to construct a new multi-attribute decision-making (MADM) method. Besides, three cases are used to demonstrate that our proposed method can overcome the disadvantages that the existing approaches cannot obtain the preference orderings of alternatives in some circumstances and involves the existence of division by zero error in the decision procedure. Compared with the two existing MADM methods, our proposed approach has the highest recognition index and the lowest error rate of division by zero. Our proposed method provides a better approach to dealing with the MADM problem in the interval-valued Fermatean fuzzy environment.


Introduction
This is a serious challenge to face up to the uncertain and fuzzy data in real-life applications involving many fields such as industry, environment science, and engineering so on. Atanassov was the first to propose the intuitionistic fuzzy sets (IFSs) [1,2]. IFSs are very efficient and useful mathematical techniques for dealing with ambiguous data [3][4][5], which are represented by the membership degree (MSD) and nonmembership degree (NMSD), and also have the restriction that 0 ≤ M S D + N M S D ≤ 1 [6]. Atanassov 1 established the notion of interval-valued intuitionistic fuzzy sets (IVIFSs) in order to overcome the constraints of prior IFS-based approaches to dealing with fuzzy data [7]. From then on, IVIFSs have been employed in a variety of contexts, including decision making [8][9][10], optimization [11][12][13], pattern recognition [14,15] and image segmentation [16]. To handle more ambiguous information, Yager et al. proposed Pythagorean fuzzy sets (PFSs), which are more adaptable than IFSs because 0 ≤ M S D 2 + N M S D 2 ≤ 1 [17,18]. To be able to flexibly enlarge or reduce the limitation range of intuitionistic fuzzy sets, Yager et al. defined q-rung orthopair fuzzy sets, their limitation is 0 ≤ M S D n + N M S D n ≤ 1 [19]. Yager et al. discovered that when q gets bigger, the universe of qualifying orthopairs expands, enabling users to process more ambiguous information [20][21][22]. When q = 3 in the q-rung orthopair fuzzy sets, Senapati and Yager called these new fuzzy sets as Fermatean fuzzy sets (FFSs) [23]. Then Senapati and Yager [24] defined the basic operation of FFSs, and gave the SCF and accuracy function of FFSs. Inspired by FFSs and IVIFSs, Jeevaraj et al. proposed the notion of interval-valued Fermatean fuzzy sets (IVFFSs) and defined some basic mathematical operations on IVFFSs [25]. The limitation of IVIFSs is 0 ≤ M S D + N M S D ≤ 1, while IVFFSs can deal with the uncertainty that 0 ≤ M S D 3 + N M S D 3 ≤ 1. Obviously, IVFFSs can handle a wider range of uncertain information than IVIFSs [26]. Therefore, IVFFS are gradually becoming a more effective mathematical model for dealing with uncertainty problems.
In recent years, numerous MADM approaches proposed in various environment, such as intuitionistic fuzzy environment [27][28][29][30]. In Pythagorean fuzzy environment [31,32], Fermatean fuzzy environment [33][34][35], interval-valued intuitionistic fuzzy environment [36,37], interval-valued Fermatean fuzzy environment [38] and other fuzzy environments [39][40][41][42][43]. In intuitionistic fuzzy environment, Senapati et al. [44] proposed some computations and some new aggregation operators, and they used these aggregation operators to address the MADM problem of the worldwide supplier selection and the health-care waste disposal method selection. In [45], Sharma et al. introduced a multi-objective bi-level chance-constrained optimization to solve a company's production planning problem in intuitionistic fuzzy environment. In Pythagorean fuzzy environment, Soltani et al. [46] developed a Pythagorean fuzzy MADM model to enhance the application process of traditional lean manufacturing. Besides, in [47], Fei and Feng introduced a dynamic multi-attribute decision framework based on PFSs that fuses decision information from different periods for addressing the effect of the time factor on MADM results. In Fermatean fuzzy environment, Saha et al. [48] extended the Delphi techniques to identify attributes based on FFSs and proposed Double Normalized MARCOS approach and constructed a novel MADM model to address the warehouse site selection. Moreover, Hezam et al. [49] presented a hybrid double normalization-based multi-aggregation approach to improve the transportation services for the person with disabilities. In the interval-valued intuitionistic fuzzy environment, Chen and Yu [50] proposed a new MADM approach based on a novel score function and the power operator. In [51], Chen et al. combined the best-worst method, decision-making trial, and evaluation laboratory to construct a MADM approach to solve resumption risk assessment amid COVID-19 in interval-valued intuitionistic fuzzy environment. Rani et al. [52] introduced Einstein aggregation operators and constructed an integrated attribute interaction via inter-criteria correlation and the complex proportional assessment approach under IVFFNs to tackle the MADM problem in interval-valued Fermatean fuzzy environment. Jeevaraj [25] created the TOPSIS approach for tackling MADM problems under the interval-valued Fermatean fuzzy environment. Rani et al. [53] devised the weighted aggregated sum product assessment approach for IVFFNs to handle the MADM problems.
However, the existing score functions in [25,[52][53][54] can not distinguish two IVFFNs in some situations. And the MADM approaches mentioned in [52,54] have the shortcomings that the MADM approach proposed in [54] and the MADM approach proposed in [52] cannot derive the preference orderings of alternatives in some situations. To solve the above problems, a novel score function and MADM approach are proposed in this paper.
In detail, our main contributions are as follows: 1. A novel score function is defined to address the shortcoming of existing methods that cannot distinguish between two IVFFNs in some situations.
In contrast to the SCF proposed by Rani and Mishra [53], Jeevaraj [25], Rani et al. [52] and Chen and Tsai [54], our proposed SCF can distinguish between any two IVFFNs. Experimental results demonstrate that our SCF has the highest differentiation rate among these above SCFs.
2. A new MADM approach has been developed based on our proposed SCF in this paper. In contrast to Rani et al.'s MADM approach [52], and Chen and Tsai's MADM approach [54], our proposed approach can obtain the preference orderings of alternatives in any situation. Three real-life cases show that our MADM method has the highest recognition index among the three MADM methods.
3. When the interval values of the non-membership degree of the IVFFNs in the decision matrix are equal to the interval values of the membership degree, our proposed MADM approach can avoid the error as the existence of division by zero while Rani et al.'s MADM approach [52] involves the existence of division by zero error in the decision procedure. Therefore, our method has the lowest error rate of division by zero as 0, which is illustrated by experiments.
This paper is arranged as follows. "Preliminaries" discusses the fundamental definitions. "Analyzing the deficiencies of the existing MADM approaches" mainly discusses the drawbacks of existing MADM approaches. In "A new MADM approach based on the proposed score function of IVFFNs and the hybrid weighted score measure of IVFFNs", a new SCF is proposed in interval-valued Fermatean fuzzy environment. Besides, the proposed novel SCF and hybrid weighted score model are used to build a new MADM approach. The new MADM approach can overcome the deficiencies of the MADM approaches described in [52,54]. "Applications of the proposed MADM method" applies three real-life cases to demonstrate the rationality and superiority of our proposed MADM method. "Comparison with the two existing MADM approaches" demonstrates the superiority of our proposed score function by differentiation rate, and we introduce recognition index and error rate to prove the higher performance of our proposed MADM method compared with existing methods. Finally, "Conclusion" draws concludes with a summary of our work and recommendations for extending our suggested MADM approach to other fuzzy environments.

Preliminaries
IVFFSs are generalization of FFSs and PFSs. The following are some basic concepts related to IVFFSs. Definition 1 [55] Let T = t 1 , t 2 , ..., t P is a finite domain of discourse. A FFS D in T is defined as follows: 3 is referred to as the degree of reluctance or indeterminacy of t to A. Definition 2 [25] Let T = t 1 , t 2 , ..., t P is a finite domain of discourse, an IVFFS F in T is defined as follows: and v F (t) represent the degree of t to F's membership and non-membership, respectively. ∀t ∈ T , μ F (t) and v F (t) are closed intervals and their lower and upper bounds are represented by μ L respectively. Hence, another equivalent expression of an IVFFS F is expressed as For each element t ∈ T , the degree of uncertainty is defined as be an IVFFN, then Rani and Mishra's SCF M of the IVFFN is depicted as follows: where M(τ ) ∈ [−1, 1]. The greater the magnitude of M(τ ), then the greater the magnitude of the IVFFN τ .
be an IVFFN, then Rani and Mishra's accuracy function H of the IVFFN is defined as follows: where H (τ ) ∈ [0, 1]. The greater the magnitude of H (τ ), then the greater the magnitude of the IVFFN τ .
be an IVFFN, then Jeevaraj's complete SCF C of the IVFFN is defined as follows: 1 2 ]. The greater the magnitude of C(τ ), then the greater the magnitude of the IVFFN τ .
be an IVFFN, then Rani et al.'s SCF R of the IVFFN is defined as follows: where R(τ ) ∈ [−1, 1]. The greater the magnitude of R(τ ), then the greater the magnitude of the IVFFN τ .
be an IVIFN, then Chen and Tsai's SCF G of the IVIFN is defined as follows: where G(τ ) ∈ [0, 2]. The greater the magnitude of G(τ ), then the greater the magnitude of the IVIFN τ .
Definition 9 [25,52,53] ) be two IVFFNs. The sorting approach between the IVFFNs F and G is defined as follows:

Analyzing the deficiencies of the existing MADM approaches
In this part, we analyze the deficiencies of existing MADM approaches in Chen and Tsai [54] based on IVIFNs and Rani et al. [52] for IVFFNs. IVFFSs is the superset of IVIFSs [25]. That is, the ranking principle on IVFFNs as generalized model still functions for IVIFNs as the subclass. Consider the following multi-attribute decision problem: P = { p 1 , p 2 , ..., p m } is a group of m plausible alternatives and R = {r 1 , r 2 , ..., r n } is a finite collection of characteristics. Consider that decision maker gives his own decision Assume the decision maker has provided the following interval-valued Fermatean fuzzy decision matrix M = ( is an evaluative IVFFN of the attribute r j relative to the alternative Following is a brief description of Chen and Tsai's MADM approach [54]. Step 1: According to Eq. (6) and the decision matrix M = ) m×n , build the score matrix SM = (σ i j ) m×n as follows: where R 1 is the set of benefit attributes and R 2 is the set of cost attributes, and G( Step 2: According to the obtained score matrix SM = (σ i j ) m×n , build the normalized score matrixS M = (σ i j ) m×n , where ≤ i ≤ m and 1 ≤ j ≤ n.
Step 5: Based on the obtained weighted normalized decision matrix W DM(η i j ) m×n , calculate the weight score W S( p i ) of alternative p i , shown as follows: where i = 1, 2, ..., m. Step where 1 ≤ i ≤ 2 and 1 ≤ j ≤ 2. [ Step 1] Based on the SCF given by Chen in Definition 8 and the decision matrix M = (χ i j ) 2×2 to construct the score matrix SM = (σ i j ) 2×2 , where r 1 and r 2 are two benefit .05, σ 12 = 0.60, σ 21 = 1.05, σ 22 = 0.60. Therefore, the score matrix can be obtained as follows: [ Step 5] Based on the weighted normalized decision matrix W DM = (η i j ) 2×2 to calculate the weight score [ Step 6] Based on the value of W S( p i ) to rank the preference orderings of alternatives, where W S( p 1 ) = 1.313 and W S( p 2 ) = 1.313. Because the weight score W S( p 1 ) = W S( p 2 ) = 1.313, the preference orderings of alternatives p 1 and p 2 acquired by Chen and Tsai's MADM approach [54] is " p 1 = p 2 ". However, the decision matrix M = (χ i j ) 2×2 indicates that the IVFFNs for alternatives p 1 and p 2 under each attribute value are different. Therefore, Chen and Tsai's MADM approach [54] has the deficiencies that this method cannot distinguish the preference orderings of alternatives and cannot help us make decisions well in this situation.
A brief description of Rani et al.'s MADM approach [52] is shown below. Consider the following multi-attribute decision problem: P = { p 1 , p 2 , ..., p m } is a group of m plausible alternatives and R = {r 1 , r 2 , ..., r n } is a finite collection of characteristics.
Step 1: According to the decision maker's opinion, con- Step 2: Based on the criteria interaction through intercriteria correlation (CRITIC) approach to obtain the weight of each attribute. Assume that Ψ = (ψ 1 , ψ 2 , ..., ψ n ) T be the group of attribute weight with ψ j ∈ [0, 1] and n j=1 ψ j = 1. The following are the calculation steps to determine the weight of attributes using the CRITIC approach: Step 2.1: According to Eq. (5) and the decision matrix Step 2.2: According to the score matrix SM = (σ i j ) m×n , build the normalized score matrixS M = (σ i j ) m×n , wherẽ and R 1 is the set of benefit attributes and R 2 is the set of cost attributes, and Step 2.3: Calculate the standard deviations ξ j of the attribute r j , where Step 2.4: Determine the relationship jk between each attribute as follows: where i = 1, 2, ..., m, j = 1, 2, ..., n and k = 1, 2, ..., n.
Step 2.6: According to the information quantity φ j of each attribute, the weight of each attribute is as follows: where i = 1, 2, ..., m and j = 1, 2, ..., n.
Step 4: Calculate the "relative degree (RD)" of each alternative.
where t is the number of benefit attribute. And the parameter θ ∈ [0, 1] represents the decision maker's strategic coefficient. The strategic parameter evaluation is as follows: if θ < 0.5, then the decision-maker displays negative behavior, and if θ > 0.5, then the decision-maker displays optimistic behavior, which indicates that the higher degree is related to the weight of benefit-type attribute, and if θ = 0.5, then the decision-maker displays neutral performance, which demonstrates that the weight values of both benefit-type and cost-type attributes are equal.
Step 5: Based on the relative degree γ i , the prioritization of each alternative is estimated. If the alternative with the highest relative degree D * , it is the optimal alternative.
Step 6: Rank the alternatives P = { p 1 , p 2 , ..., p m } based on the utility degree (UD)". The larger the value of U D( p m ), the better the preference ordering of alternative where i = 1, 2, ..., m.
Example 2 Suppose that p 1 and p 2 are two alternatives, and r 1 and r 2 are two benefit attributes, and ψ 1 and ψ 2 are the IVFFNs weight of attributes r 1 and r 2 provided by the decision maker, respectively. [ Step 1] According to the decision maker's opinion, the  2×2 , the weight of attributes can be obtained by using the CRITIC approach.
In summary, when the interval values of the non-membership degree of the IVFFNs in the decision matrix are equal to the interval values of the membership degree, Rani's MADM approach [52] fails to determine the preference orderings of the alternatives due to the division by zero error in the decision process. Therefore, Rani's MADM approach [52] has the disadvantage that the preference orderings of alternatives cannot be derived in some situations.
A new MADM approach based on the proposed score function of IVFFNs and the hybrid weighted score measure of IVFFNs In this section, firstly, a novel score function is introduced in the interval-valued Fermatean environment. And then a new MADM approach is proposed based on the novel score function and the hybrid weighted score measure to address the disadvantage of Chen and Tsai's MADM approach [54] and Rani et al.'s MADM approach [52].

The proposed score function
Following is the new proposed SCF for IVFFNs.
be an IVFFN, then the proposed SCF N of IVFFNs is defined as follows: where N (τ ) ∈ [0, 2]. The greater the magnitude of N (τ ), then the greater the magnitude of the IVFFN τ .
Proof Based on Eq. (21), then N ( , then based on Definition 9, it is obvious that It can be seen that and because μ L , then based on Definition 9, it is obvious that μ It can be seen from Definition 2, and because μ L , then based on Definition 9, it is obvious that It can be seen from Definition 2, 0 ≤ μ L It can be seen that and because μ L . Therefore, if the IVFFNs α 1 = α 2 , then N (α 1 ) = N (α 2 ).

Property 2 Let t be an IVFFN
,

The proposed MADM approach
Let r 1 , r 2 , ..., and r n be attributes, let p 1 , p 2 , ..., and p m be alternatives, and let decision matrix Figure 1 shows the process for developing the MADM approach. In addition, its steps of implementation are following.
Step 3: Based on our proposed SCF shown in Eq. (21) and the decision matrix and R 1 is the set of benefit attributes and R 2 is the set of cost attributes, and N ( Step 4: According to the normalized weightψ j and the score matrix SM = (σ i j ) m×n , mix the "Weighted Sum Model (WSM)" and the "Weighted Product Model (WPM)" to form the "Hybrid Weighted Score Model (HWM)", and then use HWM to calculate the weighted score η i for each alternative p i .
Furthermore, according to the proposed MADM approach process, computational complexity is given. In detail, the time complexity for converting each IVFFN weight into a crisp weight and normalizing each crisp weight is O(n). The time complexity for building the score matrix and calculating the weighted score for each alternative is O(mn). The last step's time complexity for ranking the alternatives is O(mlog 2 m ). Therefore, the proposed MADM approach's time complexity is O(mn)  where 1 ≤ i ≤ 2 and 1 ≤ j ≤ 2. Next, using our proposed approach to deal with the problem in Example 1, as shown in the following steps. [

Example 4 Consider the same decision matrix shown in
Our MADM method is applied to cope with the data in Example 2 as follows.
[ [ Step 2] Obtain the normalized weight according to the above crisp weight asψ 1 = 0.6447 andψ 2 = 0.3553 for attribute r 1 and r 2 , respectively. [ Step 3] Build the score matrix SM = (σ i j ) 2×2 shown as follows, by means of our proposed SCF and the above decision matrix (Here, r 1 and r 2 are benefit attributes).
Step 4] Compute the weighted score η 1 and η 2 for alternatives p 1 and p 2 by Eq. (25), according to the normalized weight and the score matrix SM = (σ i j ) 2×2 . The coefficient parameter θ is given as 0.5. Therefore, the weighted score of alternatives p 1 and p 2 can be derived, where η 1 = 0.1581 and η 2 = 0.1781. [ Step 5] Sort the alternatives p 1 and p 2 according to the weighted score η 1 = 0.1581 and η 2 = 0.1781. Since η 1 < η 2 , it is easily found that the preference orderings between alternatives p 1 and p 2 is " p 1 < p 2 ". Therefore, our new MADM method outperforms the approach in [52] in this situation. In addition, our MADM approach eliminates the possibility of not obtaining the preference orderings of the alternatives due to zero division during the operation, as shown in Example 2.
From the validations of the above two examples, the preference orderings of alternatives can be derived from our proposed MADM approach in any situation. In contrast to the MADM methods in Rani et al. [52] and Chen and Tsai [54]. It can be seen from Example 4, when MSD and NMSD have the same interval value, there is a division by 0 error in Rani et al.'s MADM process [52].

Applications of the proposed MADM method
To illustrate the proposed method's utility and superiority, three real-life applications are applied to demonstrate the proposed approach for solving MADM problems. Case 1. To demonstrate the utility of the proposed approach, it was applied to the evaluation of clean energy-driven desalination-irrigation (CEDI) systems in the Xinjiang Uygur Autonomous Region of Northwest China(modified from [56]). The Xinjiang Uygur Autonomous Region is the largest administrative division in China, with an area of more than 1.6 million square kilometers, accounting for about one-sixth of China's land area. Due to the distance from the ocean, the region forms a typical temperate continental arid climate, with an average annual precipitation of only 163.3 mm. Xin- jiang is a region that is highly dependent on irrigation due to low rainfall. Although Xinjiang lacks freshwater resources, its per capita water resources are only 3130 cubic meters, and Xinjiang has considerable potential for brackish water resources [56]. In order to select the optimal CEDI system from five different CEDI systems p i (i = 1, 2, ..., 3), expert selected four attributes to evaluate the systems. Three attributes r j ( j = 1, 2, ..., 4) relevant to the economy, agriculture, environment, and system performance were selected by consulting experts, which are r 1 : product yield, r 2 : system stability, r 3 : system adaptability, and r 4 : integrated environmental impact. In this paper, product yield (r 1 ), system stability (r 2 ), and system adaptability (r 3 ) are benefit attributes. The higher stability of the system, the better the product yield, and the better adaptability of the system, which means the better performance of the CEDI system, while the rest of the attribute is considered cost attribute.
shown as Table 1.
Input: Decision matrix M = (χ i j ) 5×4 for the evaluation of the candidates.
Output: The preference orderings of the candidates is p 1 > p 4 > p 3 > p 2 > p 5 .
In Case 1, the preference orderings of the alternatives obtained by Chen and Tsai's MADM approach [54] is p 1 > p 4 > p 3 > p 2 > p 5 . Besides, the preference orderings of the alternatives derived by Rani et al.'s MADM approach [52] is also p 1 > p 4 > p 3 > p 2 > p 5 . Obviously, our proposed approach is reasonable and practical. Case 2. A university wants to hire a talented faculty member, which is a problem in the area of human resource management (modified from [57]). The university convenes a group of decision-makers. After carefully analyzing the information, they critically assess three candidates p i (i = 1, 2, ..., 3) in these three main dimensions r j ( j = 1, 2, 3), where  is shown as Table 2.
Output: The preference orderings of the three potential investment opportunities is p 2 > p 1 > p 3 .
However, the preference orderings of the alternatives cannot derive using Rani et al.'s MADM method [52] in Case 3 because of the error of dividing by 0 in solving the MADM problem in Case 3.

Comparison with the two existing MADM approaches
In this section, firstly, we compare our proposed score function with the six existing score functions to compare the differentiation rate. And then we compare our method with existing MADM methods to demonstrate the advantages of our proposed method: (1) our proposed method has the highest recognition index compared with existing methods; (2) our proposed MADM method can avoid the division by zero error when the interval values of the non-membership degrees of the IVFFNs in the decision matrix are equal to the interval values of the membership degrees.
In conclusion, Table 4 shows whether distinguishing between different IVFFNs by six existing SCFs and our SCF on six groups, where N denotes that the score function can not distinguish two IVFFNs, Y indicates that the score function can distinguish two IVFFNs and "/" denotes that the score function is not involved in the operation. From  [52], and Chen and Tsai's [54], which in some situations cannot distinguish between two IVFFNs. That is, our SCF can identify two IVFFNs in any situations.
To further demonstrate the superiority of our proposed score function, the concept of differentiation rate is introduced as a measure of the goodness of the score function. The larger the differentiation rate, the better the score function in distinguishing any two IVFFNs. The differentiation rate (DR) is calculated as follows.

The proposed MADM method vs. two existing MADM approaches about recognition index
In order to better evaluate the performance of the MADM model, the recognition index (RI) is introduced as the evaluation index of the MADM model. The higher the recognition index of the model, the better the performance of the model. Recognition index is defined as follows: where i = 1, 2, ..., m, m is the number of alternatives and k is the number of alternatives that can be distinguished. Through Case 1, it can be seen that the the approach of Chen and Tasi [54] cannot distinguish the preference orderings of alternatives.

Our MADM method vs. the two existing approaches about error rate of division by zero
This part mainly focuses on the comparison of the proposed MADM algorithm with the two existing methods from the perspective of the error rate of division by zero. The lower the error rate of the methods, the better the performance of the models. Here the error rate of division by zero as follows: where y is the number of MADM cases, t is the number of cases that have this error of dividing by 0 when using MADM methods for processing MADM applications. When the MSD of the attribute is equal to NMSD, the method of Rani et al. suffers from the division by zero error, which leads to the inability to obtain the preference orderings of the alternatives. In Table 5, Y indicates that this MADM model has the problem of dividing by zero, and N indicates that the model does not have the problem of dividing by zero. It is clear that the ER of Rani et al.'s method is equal to 2 4 × 100% = 50%, the ER of Chen and Tsai's approach is equal to 0 4 × 100% = 0 and the ER of our proposed approach is also equal to 0 4 × 100% = 0. Therefore, our proposed method and Chen and Tsai's MADM approach do not have the error of dividing by 0.

Conclusion
In this paper, a new score function is proposed that can overcome the shortcomings of the existing score functions in Rani and Mishra [53], Jeevaraj [25], Rani et al. [52] and Chen and Tsai [54], which cannot distinguish between two IVFFNs in some situations. Based on our proposed score function, a new MADM approach is constructed that can address the defi-ciencies that (1) Some existing MADM approaches [52,54] cannot determine the preference orderings of alternatives in some situations; (2) The MADM approach in Rani et al. [52] has the disadvantage of dividing by zero in the process of decision computation, which prevents the decision-maker from obtaining the preference orderings of the alternatives.
Three practical examples illustrate the rationality and superiority of our proposed MADM method. In Case 2, our proposed method has a higher recognition index than Chen and Tsai's [54] MADM approach. Compared to the approach of Rani et al. [52], our proposed method has a lower error rate in Case 3. Therefore, our proposed MADM approach handles the MADM problem in the interval-valued Fermatean fuzzy environment more effectively.
Nevertheless, the weight of each attribute is given in advance by experts, our approach does not discuss how to determine an appropriate value. In future work, we will discuss how to determine weights, and following the idea of this paper, we can extend our proposed MADM approach to other fuzzy environments, such as hesitant Fermatean fuzzy environment [59], type-2 fuzzy environment [60][61][62] and type-3 fuzzy environment [63,64], etc.
Author Contributions Each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content.
Funding Funding was provided by the National Natural Science Foundation of China (62162055), the Gansu Provincial Natural Science Foundation of China (21JR7RA115).

Data availability statement
Enquiries about data availability should be directed to the authors.

Conflict of interest
The authors declare that they have no conflict of interest.
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